A cohomology group
of
a
$\mathbb{Z}_{2}$-orbifold model of
the symplectic
fermionic
vertex
operator
superalgebra1
Toshiyuki Abe
Ehime university
1
Introduction
In this report we calculate a cohomological group of a model ofan irrational
$C_{2}$,-cofinite simple vertex operator algebra. The cohomological group is
con-sidered by Miyamoto in a study on the category of modules for $C_{2}$,-cofinite
vertex operator algebras, and this result is just a calculation of a concrete
example. In my talk, I introduced a homology of a certain functor. But
the functor we considered is left exact. and hence the homology should be
considered as a
cohomology2.
In this report we consider the cohomologicalgroup
of the simple vertex operator algebra $SF^{+}$ which is one of examplesof irrational $C_{2}$-cofinite vertex operator algebra.
2
Preliminaries
We do not state the definition of vertex operator algebras and its
mod-ules. For them, plea,se refer to the literatures [LL], [MN] and [FHL]. Let
$(t^{\gamma}.1^{r}(\cdot., \backslash l)$
.
$1,$$\omega$) be a simplevertex operator algebra over $\mathbb{C}$, and $(M, Y(\cdot, x))$a weak $t^{r}$-module. We write
$Y(a, x)= \sum_{n\in}\underline{7}a_{(n)}x^{-n-1}$ for $a\in V$ following
[MN], where $a_{\langle n)}\in$ End $M$. We also write $L_{n}$ for the n-th mode $\omega_{(n)}$ of the
$1’\prime i?^{\backslash }asoro$ vector
$\ ’$
.
The vacuum vector 1 satisfies that for any $a\in V$ and $i\in \mathbb{Z}\geq 0\cdot a_{(i)}1=0$ and $a_{(-1)}1=a$.
A $vacuum- lik\epsilon$ vector $u\in\Lambda’I$ is a vector $u\in M$ satisfying $a_{(i)}u\cdot=0$ for a,ny $0\in t^{J}$
’
and $i\in \mathbb{Z}\geq 0$
.
We set Vac $(M)$ to be the set of all vacuum-likevectors in $\Lambda I$
.
It is known that$\backslash ^{\tau}\prime a\cdot c(\Lambda I)=$ Ker $L_{-1}=\{u\in M|L_{-1}u=0\}$
.
Actually. $L_{-1}=\omega_{(0)}$ shows that Vac$(M)\supset$ Ker $L_{-1}$
.
On the other hand if$\frac{u\in KerL_{-l},then(-ik-1)a_{\{i)}v}{1Theori\dot{g}na1tit1eisAhonuo1}=\frac{1}{k!}L_{-1}^{k}a_{(i+k)}u..\cdot Sincea_{(j)}u=0forsufficient1yogygroupa^{-}\sim- orbifo1dmode1ofthesymp1ectic$
fermionic vertex operator superalgebra.
$\underline{)}After$ my talk. Professors Matsuo and Arakawa gave me this advice. I apologize that
large positive integer$j$ and $(^{-i-1}k)\neq 0$ for any $i,$ $k\in \mathbb{Z}\geq 0$, we see that $a_{(i)}u=0$
and that $u\in Vac(M)$
.
We note that Vac $(M)$ is included in the $L_{0}$-eigenspace $AI_{0}$ of weight $0$
because $L_{0}=\omega_{(1)}$. Thus if $L_{0}$ does not have any eigenvector in $M$, then
Vac $(M)=0$.
Proposition 2.1. ([Li]) Let $u\in Vac(\Lambda l)$, and suppose that $u\neq 0$
.
Then theV-submodule
$\langle u\rangle$of
$\Lambda l$ generatedfrom
$u$ is isomorphic to V. A linear map$T”arrow\langle u\rangle$
defined
by a $|arrow a_{1-1)}u$ is a V-module isomorphism.Proof.
Let $f$ : $Varrow\langle u\rangle$ be a linear map given by $f(a)=a_{(-1)}u$ It is knownthat $\langle u\rangle$ is spanned by vectors of the form
$a_{(m)}u$ with $a\in V$ and $m\in \mathbb{Z}$
.
Since $u\in t^{7}ac$(A4), we see that $\langle u\rangle$ is in fact spanned by
$a_{(-m)}u$ with $a\in T^{\gamma}$
and $m\in \mathbb{Z}_{>0}$. Thus $f$ is surjective. We also see that $\langle u\rangle=\{a_{(-1)}u|a\in V\}$
because $(m-1)!a_{(-m)}u=(L_{-1}^{m-1}a)_{(-1)}u$ for $m\in \mathbb{Z}_{>0}$
.
Now we see $that$
$f(a_{(n)}b)=(a_{(n)}b)_{(-1)}u= \sum_{i=0}^{\infty}(\begin{array}{l}ni\end{array})(-1)^{i}(a_{(n-i)}b_{(-1+i)}u-(-1)^{n}b_{(n-1-i)}a_{(i)}u)$
$=a_{(n)}b_{(-1)}u$
$=a_{(n)}.f(b)$
for $a,$$b\in V$ and $n\in \mathbb{Z}$
.
Therefore, $f$ is a$,$
$T^{J}r$-module homomorphism. Finally
ker$f$ is a proper ideal of $V$ and hence ker$f=0$ because \dagger is simple. Thus
$f$ is a V-module isomorphism. $\square$
3
A cohomological
group
associated
to
$T/$Suppose that the adjoint module $V$ ha,s an injective resolution;
$0arrow Varrow X^{0}arrow f_{0}$
. . .
$arrow X^{n}arrow X^{n+1}f_{n}arrow^{f_{n+1}}$.
. . (exact). Then we have a cochain complex$0arrow Vac(X^{0})arrow ro$ Vac$(X^{1})arrow r_{1}$
.
$arrow\backslash Iac(X^{n})arrow r_{n}$ Vac $(X^{n+1}) \frac{r_{n+1_{1}}}{\prime}$ , where $r_{n}=f_{n}|_{P^{n}}\cdot$. We denote the corresponding cohomological group by $H(V^{\gamma})=\oplus_{n=0}^{\infty}H^{n}(T^{r})$;for $n\in \mathbb{Z}\geq 0$, where $r_{-1}=0$
.
The cohomological group is independent of the choice of injective resolutions.A vertex operator algebra $V$ is called $C_{2}$
-cofinite
if the subspace $C_{2}(V)$spanned by vectors of the form $a_{(-2)}b$ with $a,$ $b\in t/’r$ has finite codimension
in $V$
.
If $V$ is $C_{2}$-cofinite then we can show that any finitely generated weakV-module has a projective cover. Therefore, the contragredient module $V’$
has a projective resolution. In particular, $V$ has an injective resolution.
4
The
vertex operator algebra
$SF^{+}$Let $\mathfrak{h}$ be a finite dimensional vector space of dimension $2d$ with a $nonde-\wedge$
generate skew-symmetric bilinear form $\langle\cdot, \rangle$
.
Then the vector space $\mathfrak{h}=$ $\mathfrak{h}\otimes \mathbb{C}[t^{\pm 1}]\oplus \mathbb{C}K$ has a Lie super-algebra structure as follows; the even partis $\mathbb{C}K$ and the odd part is $\mathfrak{h}\otimes \mathbb{C}[t^{\pm 1}]$
.
and the super-commutation relationsare
$\{\psi\otimes t^{m}, \phi\otimes t^{n}\}=\uparrow n\langle\psi, \phi\rangle\delta_{m,-n}I\iota’’$, $[K,\mathfrak{h}]\wedge=0$
for $\phi,$$\psi\in \mathfrak{h}$ and $m,$ $n\in \mathbb{Z}$
.
Now we consider the super-algebra $\mathcal{A}$ $:=t^{\gamma}(\mathfrak{h})/\langle K-1\rangle\wedge$, where $U(\mathfrak{h})\wedge$ is the
universal enveloping algebra $of\mathfrak{h}\wedge$ and
$\langle K-1\rangle$ is the two-sided ideal of $U(\mathfrak{h})\wedge$
generated by $K-1$. Let $I\geq 0$ be the left ideal of $A$ generated by $\psi\otimes t^{n}$ for all
$\psi’\in \mathfrak{h}$ and $n\in \mathbb{Z}\geq 0$
.
We then have a left A-module $\mathcal{A}/I_{>0}$ and denote it by$SF^{3}$ It is clear that $SF$ is isomorphic to the exterior $alge\overline{b}ra\Lambda(\mathfrak{h}\otimes t^{-1}\mathbb{C}[t^{-1}])$ as vector spaces. We write $\psi_{(n)}$ for the left multiplication on $SF$ by $’\psi\otimes t^{n}$ for $\psi\in \mathfrak{h}$ and $n\in \mathbb{Z}$
.
Let 1 be the image of the unit of$\mathcal{A}$ in $SF$.
Then $SF$is spanned by vectors of the form
$\psi_{(-n_{1})}^{1_{t}}\psi_{(-n_{2})}^{2}\cdots\psi_{(-n_{\Gamma})}^{r}1$ , $(\psi^{i}\in \mathfrak{h}, n_{i}\in \mathbb{Z}>0)$
.
We define the vertex operator map $Y(\cdot, \approx)$ : $SFarrow(EndSF)[[z, z^{-1}]]$ by
$Y(1, z)=id_{T}$,
$Y( \psi_{t-1)}1, \approx)=\sum_{n\in \mathbb{Z}}\psi_{(n)})z^{-n-1}$ ,
$Y(\psi_{(-n_{1})}^{1}’\psi_{(-n_{2})}^{2}\cdots l\psi_{(-n_{f})}^{f}1, z)$
$=oO\partial^{(n_{1}-1)}Y(\psi_{t-1)}^{1}1, z)\cdots\partial^{(n_{f}-1)}Y(\psi_{t-1)}^{r}1, z)_{0}^{o}$ ,
for $’\psi’,$ $\psi^{i}\in \mathfrak{h},$ $n,,$ $n_{i}\in \mathbb{Z}_{>0}$, where $\partial^{(k)}$ $:= \frac{1}{k!}\frac{d^{k}}{d\sim- k}$ for $k\in \mathbb{Z}\geq 0$. Let $\{e^{i}, f^{i}\}_{i=1,\cdots,d}$ be a basis of $\mathfrak{h}$ satisfying
$\langle e^{i}, e^{j}\rangle=\langle f^{i}, f^{j}\rangle=0$ and $\langle e^{i}, f^{j}\rangle=-\delta_{i,j}$ for $1\leq i,j\leq d$
.
Then the Virasoro element $\omega$ is given by$\omega=\sum_{i=1}^{d}e_{(-1)}^{i}f_{(-1)}^{i}1$
.
Finally we have a, vertex operator superalgebra $(SF, Y(\cdot, z), 1,\omega)$ of central
$charge-2d$
.
The vertex operator superalgebra $SF$ has canonically an automorphism
$\theta$ defined by
$\theta(\cdot\psi_{(-n_{1})}^{1_{\text{ }}}\psi_{(-n_{2})}\cdots’\psi_{(-n_{t})})1)=(-1)^{r}’\psi_{(-n_{1})}^{1}\psi_{(-n_{2})}^{2}\cdots\cdot\psi_{(-n_{r})})^{f}1$
for any $\psi_{i}\in \mathfrak{h},$ $n_{i}\in \mathbb{Z}_{>0}$
.
The fixed point set $SF^{+}$ of $SF$ for $\theta$ is the evenpart of the vertex operator superalgebra $SF$ and the-l-eigenspac.e $SF^{-}$ is
the odd one. The even part $SF^{+}$ becomes a simple vertex operator a$,lgebra$
of central $cbarge-2d$. and $SF^{-}$ is an irreducible
$SF^{+}$-module.
5
Projective
and
injective resolutions
of
$SF^{+}$It is known that $SF^{+}$ has four irreducible modules (see [A]). These are given
by $SF^{\pm}$ and irreducible components of the unique irreducible $\theta$-twisted
SF-module. The lowest weights of$SF^{+}$ and $SF^{-}$ are $0$ and 1 respectively. Those of other two irreducible $SF^{+}$-modules are $- \frac{d}{8}$ and $\frac{4-d}{8}$
The two irreducible modules given as submodules of the irreducible $\theta-$
twisted SF-module are projective and injective. This fa,ct is not so easy but
can be shown by using the structure of Zhu’s algebra of $SF^{+}$ studied in [A].
On the other hand. $SF^{\pm}$ are not projective nor injective. Their projective
covers can be constructed as follows.
First
we consider
theSF-module
$\overline{SF}=\mathcal{A}/I>0$, where $I_{>0}$ is a left ideal of$\mathcal{A}$ genera,ted by $\psi t_{\vee^{\backslash }}^{-}:\cdot’$)$t^{n}$ with $\psi\in \mathfrak{h}$ and $n\underline{\in}\mathbb{Z}_{>0}$
.
We see that$\overline{SF}$
is generated
from the vector $\wedge 1=1+I_{>0}$ and that $SF\cong\Lambda(\mathfrak{h}_{-}1_{\backslash }^{\backslash },-/I\mathbb{C}[t^{-1}])$ as vector spaces.
We define the action of $\theta$ on
$\hat{T}$ by
$-\pm$
for a,ny $\psi_{i}$) $\in \mathfrak{h},$$n_{i}\in \mathbb{Z}\geq 0$
.
We denote by $SF$ by $the\pm 1$-eigenspace for $\theta$. We$-\pm$ $-\pm$
note tha,$t$ they are $SF^{+}$-modules and $(SF )’\cong SF$ respectively. We use
the following conjecture.
Conjecture. The $S$F-modules $\overline{SF}^{\pm}$ are projective and injective. $-\pm$
Assuming this conjecture is true. we can find that $SF$ are projective
covers of the $\llcorner\sigma\prime F^{+}$-module $SF^{\pm}$ respectively as follows. By construction, we
have a,n SF-module epimorphism $\phi_{0}$ : $\overline{SF}arrow SF$ defined by $\phi_{0}(\psi_{(-n_{1})}^{1}\psi_{(-n_{2})}^{2}. . . ’\psi_{(-n_{\Gamma})}r^{\wedge}1)=\psi_{(-n_{1})}^{1}\psi_{(-n_{2})}^{2}\cdots\psi_{(-n_{r})}^{r}1$
for $\iota i_{i}\in \mathfrak{h},$ $n_{i}\in \mathbb{Z}\geq 0$
.
By definition $\phi_{0}$ gives epimorphisms$\overline{SF}^{\pm}arrow SF^{\pm}$
respectively. We set $M_{0}^{\prime^{r}}=$ ker$\phi_{0}$
.
Then $W^{r_{0}}$ is an SF-submodule of $\hat{S}F$generated from $e_{(0)}^{i}1\wedge$ and $f_{(0)}^{i}1\wedge$ for 1 $\leq i\leq d$
.
We also see that $J\psi_{0}^{r}=$$-+$ — $-\pm$
$(Tt_{0}\cap SF)\oplus(\iota r_{0\cap SF}^{J}" )$ and the submodules $\mathfrak{j}fl_{0}^{7}/\cap SF$ are indecomposable.
$-\pm$
Hence $SF$ are projective covers of $SF^{\pm}$ respectively.
We now state that $SF$ has the following projective resolution.
Theorem 5.1. $Th,eSF^{+}$-module $SF$ has a projective resolution
.
.
. $arrow P^{n+1}arrow P^{n}arrow\cdotsarrow P^{0}arrow SFarrow 0$,with $P^{n}=SF$$-\oplus h(n+1)$; the direct sum
of
$h(n)$-copiesof
$\overline{SF}$.
The number $h(n)$ is given as follows: Let
$44=(\begin{array}{lllll}0 0 0 (^{2d})(^{2^{0}d})(_{2}^{2^{l}d})-1 0 0 \vdots 0 -1 0 \vdots\vdots \ddots \vdots \vdots 0 0 -l (_{2d-l}2d)\end{array})$
be a $2d\cross 2d$-matrix, and set
Then $h(n)$ is the $2d$-th component of $t$)$(r)$
.
Hence$h(1)=1$ , $h(2)=2d$, $h(3)=d(2d+1)$, $\cdot$
..
.
In the case $d=1$, we have $d(n)=n$
.
Since $\overline{SF}’$
, the contragredient mcdule to $\overline{SF}$
, is isomorphic to $\overline{SF}$
, by this
theorem, we have an injective resolution
$0arrow SFarrow P^{0}arrow P^{0}arrow\cdotsarrow P^{n}arrow\cdots$
By studying the structure of $\overline{SF}$
in detail, we get
Theorem 5.2. The irreducible $SF^{+}$-modules $SF^{\pm}$ have injective resolutions
$0arrow SF^{\pm}arrow P^{0,\pm}arrow\cdotsarrow P^{n,\pm}arrow P^{n+1,\pm}arrow\cdots$
respectively, $wh.e\uparrow^{\backslash }\epsilon$
$P^{n,\pm}=\{\begin{array}{ll}- (SF )^{\oplus h(n+1)} if n is even:(\downarrow\overline{SF} )^{\oplus h(n+1)} if n is odd.\end{array}$
6
Cohomological
group
$H^{\bullet}(SF^{+})$By Theorem 5.2, we get tbe cochain complex
$0arrow Vac(P^{0,+})arrow r0$ Vac $(P^{1.+})arrow^{1}f$
.
..
$arrow Vac(P^{n,+})arrow r_{n}$ Vac$(P^{n+1,+}) \frac{r_{n+1_{t}}}{r}$Wenotethat Vac. $(\overline{SF}^{+})=\mathbb{C}e_{0}^{1}\cdots\epsilon_{(0)}^{d}f_{(0)}^{1}\cdots f_{(0)}^{d}i$ and Vac $(\overline{SF}^{-})=0$
.
Hence Vac $(P^{n,+})\cong\{\begin{array}{ll}\mathbb{C}^{h(n+1)} if n is even,0 if n is odd.\end{array}$for $n\geq 1$
.
We ca,$n$ observe that${\rm Im} r_{n}=0$ for $n\in \mathbb{Z}\geq 0$,
ker$r_{n}=\{\begin{array}{ll}Vac (P^{n,+}) if n is even,0 if_{7?}\cdot is odd.\end{array}$
Theorem 6.1.
$H^{i}(SF^{+})\cong \mathbb{C}^{h(i+1)}$
if
$i$ is even,$H^{i}(SF^{+})=0$
if
$i$ is odd.Remark 6.2. We can also define $H^{i}(SF^{-})$
.
Then we have $H^{i}(SF^{-})\cong 0$ if $i$ is even and $H^{i}(SF^{-})\cong \mathbb{C}^{h(i+1)}$ if $i$ is odd.7
A
projective
resolution in
the
case
$d=1$We explain the projective resolution of $l_{\vee}qF$ in the case $d=1$
.
For simplicity,we set $\epsilon=e^{1}$ and $f=f^{1}$. In this case, the submodule ker$\phi_{0}=T\phi_{0}^{7}$ is
generated by $e_{(0)}i$ and $f_{(0)}\hat{1}$, and the submodule generated from $e_{(0)}.f_{(0)}1\wedge$ is
isomorphic to $SF$ because $e_{(0)}.f_{(0)}1\wedge$ is a vacuum-like vector. Therefore, we
have the following sequence of submodules;
$0\subset SF\subset|V_{0}\subset\overline{SF}$
.
One sees that $\overline{SF}/M_{0}^{r}\cong SF$ and $\nu v_{0}^{7}/SF\cong SF\oplus SF$.
Now we consider the SF-module epimorphism $\phi_{1}$ : $\overline{SF}\oplus\overline{SF}arrow T\phi_{0}^{r}$
defined by
$\phi_{1}(u1’.\iota 1)=u\epsilon_{(0)}1+t)f_{(0)}1\wedge\wedge\wedge\wedge$,
where $u_{:}v\in\Lambda(\mathfrak{h}\otimes \mathbb{C}[t^{-1}])$. Then we see that the kernel of $\phi_{1}$, denoted
by $f\phi_{1}^{7}$, is the $SF$-submodule of
$\overline{SF}^{\oplus 2}$
generated
by the vectors $(e_{(0)}10)\wedge,$,$(f_{(0)}1, e_{(0)}1)$ and $(0.f_{(0)}1)\wedge$
.
If we draw an extension of $X$ by $l’-$ as
$X$
$Y\downarrow$
then we have the following pictures;
$SF$
$\swarrow$ $\searrow$
$\overline{SF}=$ $SF$ $SF$
$\searrow$ $\swarrow$
and
$SF$
垣\acute ’0 $=$ $\swarrow$ $\searrow$
$SF$ $SF$
We also see that
$SF$ $SF$
$\swarrow$ $\searrow$ $\swarrow$ $\searrow$
$-\oplus 2$
$SF$ $=SF$ $SF$ $SF$ $SF$,
$\searrow$ $\swarrow$ $\searrow$ $\swarrow$
$SF$ $SF$
and
$SF$ $SF$
垣\mbox{\boldmath$\tau$}1 $=$ $\swarrow$ $\searrow$ $\swarrow$ $\searrow$
$SF$ $SF$ $SF$
By the same way, for $n\in \mathbb{Z}_{>0}$, we consider a SF-module homomorphism
$-\oplus n$ $-\oplus(n-1)$
$\phi_{n-1}$ : $SF$ $arrow SF$ defined by
$\phi_{n-1}(u^{1}1..u^{n}1)\wedge,.,\wedge$
$=(u^{1}e_{(0)}1\wedge+u^{2}.f_{(0)}^{\wedge}1, u^{2^{\wedge}}\epsilon_{(0)}1+u^{3^{\wedge}}f_{(0)}1, \ldots, u^{n-1^{\wedge}}e_{(0)}1+u^{n}f_{(0)}1)\wedge$
with $u^{1},$
$\ldots,$ $u^{n}\in\Lambda(\mathfrak{h}\otimes \mathbb{C}[t^{-1}])$
.
Then we can show that${\rm Im}\phi_{n}=$ ker $\phi_{n-1}$
for $7l\in \mathbb{Z}_{>0}$ and we have the exact sequence
. . . $arrow^{\phi_{n+1}}\overline{SF}^{\oplus(n+1)}arrow\phi_{n}\overline{- S^{l}F}^{\oplus n}arrow^{\phi_{\mathfrak{n}-1}}$.
.
.
$arrow\phi_{1}\overline{SF}-\phi_{0}SFarrow 0$.
We recall the action of $\theta$ on $\overline{SF}$
.
We extend the action of $\theta$ to tha,$t$ on$-\oplus n$
$SF$ with diagonal action. Then it is easy to see that $\theta 0\phi_{n}0\theta=-\phi_{n}$ for any $n\in \mathbb{Z}\geq 0$
.
Therefore, the projective resolution above gives rise to twoprojective resolutions
$-\epsilon^{\pm}$ $-\cdot\pm$
. .
.
$arrow^{\phi_{n+1}}(SF^{n\cdot+1})^{\Phi(n+1)}arrow\phi_{n}(SF^{rightarrow n})^{\oplus n}arrow^{\phi_{n-1}}$. .
..
.
.
$arrow(\overline{SF}^{\mp})^{\oplus 2}arrow\phi_{1}\overline{\vee 9F}^{\pm}arrow,-\phi_{0}9F^{\pm}arrow 0$ ,where $\vee^{\pm}cn$ is defined by
$\epsilon_{n}^{\pm}=\{\begin{array}{ll}\mp if n is even\pm if n is odd.\end{array}$
References
[A] SF. Abe, A $\mathbb{Z}_{2}$-orbifold model of the symplectic fermionic vertex
operator superalgebra, Math. Z., online.
[FHL] I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches
to vertex operator algebras and modules, $\Lambda\cdot Iem$
.
Amer. Math. Soc.104 (1993).
[Li] H.-S. Li, Symmetric invariant bilinear forms on vertex operator
al-gebra.$s_{J}$
.
J. Pure. and Appl. Algebra 96, Issue 3 (1994), 279-297.[LL] H.-S. Li and J. Lepowsky, Introduction to vertex operator algebms
and their representations, Prog. Math., Birkh\"auser, 2004.
[MN] A. Matsuo and K. Nagatomo, Axioms for a Vertex Algebra and
the Locality of Quantum Fields, MSJ Memoirs 4, Mathematical
